What is the integral of x^2e^{1/x} ?

The integral of x^2e^{1/x} is 1/3 e^{1/x}x^3+1/6 e^{1/x}x^2+1/6 e^{1/x}x-1/6 \Ei(1/x)+C

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integral of x^2e^{1/x}

\int x^{2} e^{\frac{1}{x}} d x

\frac{1}{3} e^{\frac{1}{x}} x^{3} + \frac{1}{6} e^{\frac{1}{x}} x^{2} + \frac{1}{6} e^{\frac{1}{x}} x - \frac{1}{6} Ei (\frac{1}{x}) + C

Solution steps

\int x^{2} e^{\frac{1}{x}} d x

Apply u-substitution

: \int - \frac{e ^{u}}{u ^{4}} d u

= \int - \frac{e ^{u}}{u ^{4}} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int \frac{e ^{u}}{u ^{4}} d u

Apply Integration By Parts

: - \frac{e ^{u}}{3 u ^{3}} - \int - \frac{e ^{u}}{3 u ^{3}} d u

= - (- \frac{e ^{u}}{3 u ^{3}} - \int - \frac{e ^{u}}{3 u ^{3}} d u)

\int - \frac{e ^{u}}{3 u ^{3}} d u = - \frac{1}{3} (- \frac{e ^{u}}{2 u ^{2}} + \frac{1}{2} (- \frac{e ^{u}}{u} + Ei (u)))

= - (- \frac{e ^{u}}{3 u ^{3}} - (- \frac{1}{3} (- \frac{e ^{u}}{2 u ^{2}} + \frac{1}{2} (- \frac{e ^{u}}{u} + Ei (u)))))

Substitute back

u = \frac{1}{x}

= - (- \frac{e ^{\frac{1}{x}}}{3 ( \frac{1}{x} ) ^{3}} - (- \frac{1}{3} (- \frac{e ^{\frac{1}{x}}}{2 ( \frac{1}{x} ) ^{2}} + \frac{1}{2} (- \frac{e ^{\frac{1}{x}}}{\frac{1}{x}} + Ei (\frac{1}{x})))))

Simplify

- (- \frac{e ^{\frac{1}{x}}}{3 ( \frac{1}{x} ) ^{3}} - (- \frac{1}{3} (- \frac{e ^{\frac{1}{x}}}{2 ( \frac{1}{x} ) ^{2}} + \frac{1}{2} (- \frac{e ^{\frac{1}{x}}}{\frac{1}{x}} + Ei (\frac{1}{x}))))) : \frac{1}{3} e^{\frac{1}{x}} x^{3} + \frac{1}{6} e^{\frac{1}{x}} x^{2} + \frac{1}{6} e^{\frac{1}{x}} x - \frac{1}{6} Ei (\frac{1}{x})

= \frac{1}{3} e^{\frac{1}{x}} x^{3} + \frac{1}{6} e^{\frac{1}{x}} x^{2} + \frac{1}{6} e^{\frac{1}{x}} x - \frac{1}{6} Ei (\frac{1}{x})

Add a constant to the solution

= \frac{1}{3} e^{\frac{1}{x}} x^{3} + \frac{1}{6} e^{\frac{1}{x}} x^{2} + \frac{1}{6} e^{\frac{1}{x}} x - \frac{1}{6} Ei (\frac{1}{x}) + C

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What is the integral of x^2e^{1/x} ?
The integral of x^2e^{1/x} is 1/3 e^{1/x}x^3+1/6 e^{1/x}x^2+1/6 e^{1/x}x-1/6 \Ei(1/x)+C